Both roots of the quadratic equation $x^2 - 63x + k = 0$ are prime numbers. Find the number of possible values of $k.$
Let $p$ and $q$ be the roots.  Then by Vieta's formulas, $p + q = 63.$

If both $p$ and $q$ are odd, then $p + q$ is even, so one of $p$ or $q$ must be even.  This means one of $p$ and $q$ is 2, and the other is $63 - 2 = 61.$  Therefore, $k = 2 \cdot 61 = 122,$ so there is only $\boxed{1}$ possible value of $k.$